Finite Group Databasebeta

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#72283544

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#13674
commutator, Frattini, cosocleC2×#492C24
prime radical#72283498C2
LCSG ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42 ◁ C22×#492 ◁ G
LECSG ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

7459 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 7459 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 64

Polycyclic presentation:

2^11|b^2=hijk,c^2=hijk,d^2=e,f^2=ik,h^2=j,i^2=k,d^a=de,d^b=dg,d^c=df,f^b=fh,f^d=fi,g^b=gik,g^c=ghi,g^f=gi,h^b=hj,h^d=hi,h^g=hj,i^b=ij,i^d=ik,i^g=ik,j^d=jk

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 3, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);